Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #25443 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
| direct product | C2×#28506 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3154 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3084 conjugacy classes of subgroups.
| minimal degree | μ(G) = 38 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 38 points:
(1,2);(3,4,5,6)(7,8)(9,15)(10,22)(11,14)(12,17)(13,18)(16,24)(19,27)(20,34)(21,29)(23,28)(25,32)(26,31)(30,33)(35,38)(36,37);(8,14,18,17)(9,19,21,26)(10,23)(15,29)(16,22,32,28)(20,35,36,33)(24,25)(27,31);(4,6)(7,9)(8,15)(10,20)(11,19)(12,26)(13,21)(14,27)(16,30)(17,31)(18,29)(22,34)(23,36)(24,33)(25,35)(28,37)(32,38);(7,10,11,24,13,23,12,25)(8,16,17,22,18,32,14,28)(9,20,19,33,21,36,26,35)(15,30,31,34,29,38,27,37)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6372873);